Student asked me what's the transformations

a student online sent me this video for me to be able to identify the Transformations so what I want to do in this video is to help you as well as help them with their problem so let's go ahead and write out the problem on this board so therefore I can kind of work through it all right so if I was approaching this problem and helping the student they're sitting right next to me a couple things I would recognize or a couple things I would work out with them first we need to understand when we're trying to identify the Transformations we want a quadratic to be in the form y = a * x - h^ 2 + K that is going to be our vertex form where a remember is going to be our stretch as well as our compression vertical stretch and compression also a is going to reflect about the x-axis H is going to be your horizontal shift left and right and K is going to be your vertical shift up and down the problem that we have in this equation is this is in standard form y = ax^2 + BX plus C so what we need to do in this problem before we can identify the Transformations we need to go from this form all the way over to that form and the process of doing that is called completing the square now to go about completing the square there's a couple things we need to kind of look at first or kind of have like a little checklist the first thing is we cannot have a coefficient of our quadratic term so what we need to do is we need to be able to get that term away from the X squ now the way to do that is going to be a little confusing in this problem because a lot of times students don't think to apply this process but I don't want to kind of do it a different way I want to be able to approach everything thing we're keeping the Y on the left hand side and we're just going to only work here on this right hand side so to get this -2 outside of it what I simply need to do is I need to factor out a -2 from the -2X 2 as well as from the 5x and that's even more confusing because I know I can factor a -2 from here but how do I factor out a -2 from the five well here's the kind of important things if you have a five and let's say you factor out a -2 well then all you need to Simply do is rewrite A5 divided by -2 and you can see that this still equals A5 so when I'm factoring out a-2 think about that as like dividing out the -2 and since -2 doesn't evenly divide into five my resulting answer is going to be 5 over -2 so the answer or the expression is going to look like this all right so now what I've done is I have now taken out this factor out this ne-2 so now inside this parentheses I have an x^2 minus a 5 x now I have something I can work with for completing the square because again the idea of completing the square is I need to go from standard form where I have this X squ all the way over to now where I have a quantity x - h^2 so what we need to do is we need to create what we call a perfect square trinomials because perfect square trinomials can be factored down into binomial squared if you remember like a common one like x^2 + 2x + 1 this is called a perfect square trinomial this can be Factor down into x +1 * an x + 1 which can be simplified into an x +1 quantity s that is what we need to create unfortunately this has a a middle term of two this has a middle term of -5 so it's going to be a little bit more confusing to be able to create this value C or is going to create the perfect square trinomial but thankfully for us there is a formula for us to be able to identify c c is equal to B / 2 co^ s so what I'm simply going to do is take the5 and you could even do positive because it doesn't matter you're going to square it 55s divid two Quin Square well remember guys dividing by two is the same thing as multiplying by 1/2 so when I multiply by 1/2 on the top and the bottom that's going to go to one I'm going to be left with a -5 over 4^ 2ar which is going to give me a 25 over 16 now this value is what we're going to plug inside this parentheses so y = a -2 * x^2 - 5 X plus a 25 over 16 but here's the little issue we can't just randomly add a 25 over 16 onto the right side of the equation right remember whatever you do on one side you have to do on the other side but at the beginning of the video I told you I don't want to do anything on this left hand side I want to keep all my work on the right hand side so to keep this equation still equal if I add a 25 over 16 then I'm going to have to subtract a 25 over 16 right as long as you add and subtract the same value on the same side of the equation you're still keeping the equation equivalent but here's the thing I actually didn't add a 25 or 16 because remember the dist this is inside the parentheses that's being multiplied byga -2 that means everything inside here is being multiplied by -2 so what I actually added was a 25 over 16 * a -2 so when I subtract a 25 or 16 I also need to multiply that by a -2 now what I simply I'm going to look into doing is I need to create this binomial squared all right so how do we go about factoring this to a binomial squared this was not a problem a difficult problem to go and Factor right X2 plus 2X plus one what two numbers multi give you one add to give you two X+ 1 * X+ 1 X+ One S but what do you do when you have something like fractions well thankfully there's actually another easy way for us to be able to do this all we simply need to look into is our B / two once it's simplified so our B / two is going to be our value that's going to give that's going to be in our binomial square if your middle term is negative that means your binomial is going negative so therefore I have a x minus a 54 quantity squ and then over here I just need to go ahead and simplify this well a negative * a negative is going to be a positive and this again reduce to a 1 over 8 so therefore now I'll have a positive 25 over 8 okay so now we've gone from standard form all the way to vertex form now we just need to go ahead and write out our Transformations but for the sake of this video I'm not going to write them out but I will verbally say them we're going to have a reflection about the x-axis so instead of the graph opening up it's going to be opening down since my a is larger than one two is larger than one I'm going to have a vertical stretch as a factor of two it's x - 54 remember the formula guys is X - H that means H is equal to a 54 right it's xus 54 so H equal to 54s that means I'm going to shift the graphs 54s over to the right and my K is going to be a postive 25 over 8 so for K that means I'm going to shift the graph up 25 over 8 units which looks like it's going to be 3 * 8 so 3 and 1/8 would roughly be there but there you go ladies and gentlemen that is the process for finding the uh Transformations I hope that helped the student and I hope it's going to help you but